Jet Propulsion/1D Analysis

Ramjet

The thrust produced by a ramjet is

$T={\dot {m}}u_{inlet}({\sqrt {\tau _{b}}}-1)$

where τ_b is the total temperature ratio produced by the combustor. Metallurgy and the availability of cooling will limit the maximum temperature that can be sustained in the combustor. We can define τ_max as the maximum total temperature ratio compared to the inlet conditions and τ_c as the temperature ratio relating the static and total temperatures. Then

$\tau _{max}=\tau _{b}\tau _{c}=\tau _{b}\left(1+{\frac {\gamma -1}{2}}M_{0}^{2}\right)$

Thus as τ_c increases with speed for a fixed maximum temperature τ_max stays constant and τ_b must reduce. If the theoretical frontal area of the ramjet is constant then the mass flow through the ramjet will increase linearly with the Mach number. At the same time the heat added diminishes. The thrust then is:

$T=A\rho (M_{0}a_{0})^{2}\left[{\sqrt {\frac {\tau _{max}}{1+{\frac {\gamma -1}{2}}M_{0}^{2}}}}-1\right]$

When τ_max=τ_c the term in brackets goes to zero and the thrust vanishes. The thrust for a given τ_max is shown in the following figure (the dashed line is the peak thrust):